Cross-coefficient

The coefficient Bij characterizes a bimolecular interaction between molecules i and J, and therefore Bij = Bji. Two lands of second virial coefficient arise Bn and By, wherein the subscripts are the same (i =j) and Bij, wherein they are different (i j). The first is a virial coefficient for a pure species the second is a mixture property, called a cross coefficient. Similarly for the third virial coefficients Cm, Cjjj, and are for the pure species and Qy = Cyi = Cjn, and so on, are cross coefficients. 

Values for the cross coefficients and their derivatives in these equations are provided by writing Eq. (4-203) in extended form ... 

The sign of the cross coefficient Lv< determines the direction of the electroosmotic flux and of the cation flux. From Eq. (6.2.12) we have... 

By definition, Lnjl > 0. The sign of Js is determined by the sign of the cross coefficient Lvjr and its absolute value. If Lv <0, the volume flux of the solvent occurs in the direction from more dilute to more concentrated solutions (i.e. in the direction of the osmotic pressure gradient). If LVJt is smaller than then the solute flows in the direction of the drop of... 

Tsonopoulous, C. "Second Virial Cross Coefficients Correlation and Prediction of k..," Adv. in Chem., in press(1979). 

The cross coefficients are contained in the electro-osmotic drag coefficients ... 

For inter diffusion between same-valence ions (ionic exchange) in an aqueous solution, or a melt, or a solid solution such as olivine (Fe +, Mg +)2Si04, an equation similar to Equation 3-135c has been derived from the Nemst-Planck equations first by Helfferich and Plesset (1958) and then with refinement by Barter et al. (1963) with the assumption that (i) the matrix (or solvent) concentration does not vary and (ii) cross-coefficient Lab (phenomenological coefficient in Equation 3-96a) is negligible, which is similar to the activity-based effective binary diffusion treatment. The equation takes the following form ... 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

The subscripted coefficients, B, C. .., are functions of T only, and their numerical values are unchanged on permutation of the subscripts. Coefficients having identical subscripts, eg, Bn and C222, are properties of pure species those having mixed subscripts, eg, Bl2 = B2l, C122 = C212 = C221, are mixture properties and are called interaction or cross-coefficients. 

The Onsager reciprocity relation, when applied to the present context, predicts that the cross coefficients a2 and a, which determine the rate of flow ofliquid due to the applied electric field and the current passing due to a hydrostatic pressure difference, respectively, are equal, Le.,... 

Equation (5.56) relates the correlation factor fA with the cross coefficient LAA . From the Nernst-Einstein relation we know that LAA = bA-cA = DAcA/R T. For a tracer experiment with a negligible fraction of A, the jump conservation requires that Da = Dv-Nv, so that instead of Eqn. (5.56) we have... 

For very dilute solid solutions of B in A, the basic physics of diffusional mixing is the same as for (A, A ). An encounter between VA and BA is necessary to render the B atoms mobile. But B will alter the jump frequencies of V in its surroundings and therefore numerical values of the correlation factor and cross coefficient are different from those of tracer A diffusion. Since the jump frequency changes also involve solvent A atoms, in addition to fB, the numerical value of fA must be reconsidered (see next section). 

The Darken-type equation (D = [NB-DA + NA-DB,]-flh, see Section4.3.3) is obtained only if cross coefficients are zero. In order to evaluate these cross coefficients, kinetic theory beyond the phenomenological approach is needed [A. R. Allnatt, A. B. Lidiard (1993)]. 

Let us now turn to cross effects proper between electronic and ionic fluxes. Considering the general nature of cross effects in crystals, our analysis will be performed in some depth. It gives us the tools for a correct application of SE transport theory (see Section 4.2.2) and explains to some extent the physical meaning of the cross coefficients. Let us illustrate the problem using a semiconducting binary compound such as a transition-metal oxide. In A, 0 crystals with the B1 structure, oxygen... 

Equations (8.48)-(8.50) define three independent transport coefficients for the two building units (A,h), namely L, and Lhh, in terms of the 21 independent transport coefficients of the SE set. They are sufficient to describe the transport in A O. The cross coefficient LAh expresses the coupling between the ionic and electronic fluxes. If ATh ) = 0, the electronic flux is due only to the cross effect and given by... 

This experimental verification entitles us to apply the theory of linear energy converters to oxidative phosphorylation.2 Kedem and Caplan have introduced a useful normalization of the straight and the cross coefficients of the scheme [equations (1) and (2)]. 

As more experimental data have been obtained, empirical correlations have been suggested for determining the values of the cross coefficients, such as ai, from the properties of the mixtures rather than from those of the pure components. 

If the cross coefficients are neglected in equation 5.116 [115] then, for codiffusion, where the species, A and B, are diffusing in parallel, two diffusivities, DAA DA, and, DBB DB, can be defined [114], Thus, to follow the sorption kinetics, during the codiffusion experiment, it is necessary to track both components in the mixture [125], However, in the case of counterdiffusion, where molecule, A, moves in and molecule, B, moves out , one effective diffusivity De = DA DB can be defined to describe the process [90,125], Thus, to track the sorption kinetics, during the counterdiffusion experiment, it is necessary to follow only of one of the components in the mixture. Consequently, to measure these diffusivities equation 5.107 was applied. In the present cases, D, was substituted in equation 5.107 by De, for counterdiffusion. 

Values of the pure-species virial coefficients Bj, B, etc., can be determin from the generalized correlation represented by Eqs, (3.47) through (3.49). cross coefficients Bik, By, etc., are found from an extension of the same correlatic For this purpose, Prausnitzf has rewritten Eq. (3.47) in the more general for... 

Here CiU and C222 are the third virial coefficients for pure species 1 and 2 whereas C,12 and C122 are cross-coefficients. Published generalized correlation for third virial coefficients ) are based on a very limited supply of experimen data. Consistent with the mixing rules of Eq. (11.44) and (14.1), the temperatu ( derivatives of B and C are given exactly by... 

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

These equations are called the phenomenological equations, which are capable of describing multiflow systems and the induced effects of the nonconjugate forces on a flow. Generally, any force Xt can produce any flow./, when the cross coefficients are nonzero. Equation (3.175) assumes that the induced flows are also a linear function of non-conjugated forces. For example, ionic diffusion in an aqueous solution may be related to concentration, temperature, and the imposed electromotive force. 

This relation suggests that the local rate of entropy production is a quadratic form in all forces and in all flows if the cross coefficients differ from zero. 

Where A1 and A2 are the affinities for reactions 1 and 2. The linear reaction flows with vanishing cross coefficients are... 

Similar equations can be written for components J2 and J3. The coefficients Dx x and D22 are the main coefficients they are not self-diffusion coefficients. Du and D2l are the cross-coefficients and assumed to be equal to each other for binary gas mixtures. 

If nitrogen is used instead of helium, then the values of Z)13 and D23 approach Du. Therefore, matrix [ko] becomes a diagonal matrix as the cross-coefficients ky and ky vanish. This is called the pseudo-binary case, and we have N] = —N2. 

For a binary fluid at mechanical equilibrium and for diffusion based on the mass-average velocity, we can now establish a set of phenomenological equations (Eqs. 7.6 and 7.7) with nonvanishing cross coefficients, and hence represent the coupled heat and mass flows... 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

Concentration effects on the heats of transport and the thermal diffusion ratio of chloroform with various alkanes at 30°C and 1 atm are seen in Table 7.6. Table 7.7 shows the experimental heats of transport at various concentrations and at temperatures 298 and 308 K for binary mixtures of toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm. The absolute values of heats of transport decrease gradually as the concentrations of the alkane increase. Table 7.7 also contains values of cross coefficients obtained from easily measurable quantities and the thermodynamic factor. 

Coefficients Lqq and Lt] are associated with the thermal conductivity k and the mutual dififusivity D, respectively, while the cross coefficients Liq and Lqi define the coupling. Thermal conductivity (k) is related to Lqq by k = LqJT, while the thermal diffusion coefficient is related to Liq by Liq = pDTi. Tables 7.9 and 7.10 show the values of the phenomenological cofficients Lq for the ternary mixture of toluene (l)-chlorobenzene (2)-bromobenzene (3) at 298.15 and 308.15 K. 

These equations show the relationships between the degrees of coupling and the cross coefficients Lqi. 

Here, the cross coefficients are eliminated by using the heats of transport. These equations may be solved by using appropriate initial and boundary conditions in Eq. (7.105). 

This equation is called the relative thermoelectric power of the metal a against b . Since the transfer of entropy depends on the cross coefficients T12 or A2I, this derivation represents coupling between the electrical and thermal phenomena. 

The linear phenomenological reaction flows with vanishing cross-coefficients are... 

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